Alain Albouy
77, avenue DenfertRochereau, 75014, Paris
Institut de Mécanique Céleste et de Calcul des Éphémérides, CNRS
Publications:
Albouy A., Zhao L.
Darboux Inversions of the Kepler Problem
2022, vol. 27, no. 3, pp. 253280
Abstract
While extending a famous problem asked and solved by Bertrand in 1873, Darboux
found in 1877 a family of abstract surfaces of revolution, each endowed with a force function,
with the striking property that all the orbits are periodic on open sets of the phase space.
We give a description of this family which explains why they have this property: they are
the Darboux inverses of the Kepler problem on constant curvature surfaces. What we call the
Darboux inverse was briefly introduced by Darboux in 1889 as an alternative approach to the
conformal maps that Goursat had just described.

Albouy A.
Projective Dynamics and First Integrals
2015, vol. 20, no. 3, pp. 247276
Abstract
We present the theory of tensors with Young tableau symmetry as an efficient computational tool in dealing with the polynomial first integrals of a natural system in classical mechanics. We relate a special kind of such first integrals, already studied by Lundmark, to Beltrami’s theorem about projectively flat Riemannian manifolds. We set the ground for a new and simple theory of the integrable systems having only quadratic first integrals. This theory begins with two centered quadrics related by central projection, each quadric being a model of a space of constant curvature. Finally, we present an extension of these models to the case of degenerate quadratic forms.

Albouy A.
Projective dynamics and classical gravitation
2008, vol. 13, no. 6, pp. 525542
Abstract
We show that there exists a projective dynamics of a particle. It underlies intrinsically the classical particle dynamics as projective geometry underlies Euclidean geometry. In classical particle dynamics a particle moves in the Euclidean space subjected to a potential. In projective dynamics the position space has only the local structure of the real projective space. The particle is subjected to a field of projective forces. A projective force is not an element of the tangent bundle to the position space, but of some fibre bundle isomorphic to the tangent bundle.
These statements are direct consequences of Appell’s remarks on the homography in mechanics, and are compatible with similar statements due to Tabachnikov concerning projective billiards. When we study Euclidean geometry we meet some particular properties that we recognize as projective properties. The same is true for the dynamics of a particle. We show that two properties in classical particle dynamics are projective properties. The fact that the Keplerian orbits close after one turn is a consequence of a more general projective statement. The fact that the fields of gravitational forces are divergence free is a projective property of these fields. 
Albouy A., Fu Y.
Euler Configurations and QuasiPolynomial Systems
2007, vol. 12, no. 1, pp. 3955
Abstract
Consider the problem of three point vortices (also called Helmholtz' vortices) on a plane, with arbitrarily given vorticities. The interaction between vortices is proportional to $1/ r$, where $r$ is the distance between two vortices. The problem has 2 equilateral and at most 3 collinear normalized relative equilibria. This 3 is the optimal upper bound. Our main result is that the above standard statements remain unchanged if we consider an interaction proportional to $r^b$, for any $b<0$. For $0 < b < 1$, the optimal upper bound becomes 5. For positive vorticities and any $b<1$, there are exactly 3 collinear normalized relative equilibria. The case $b=2$ of this last statement is the wellknown theorem due to Euler: in the Newtonian 3body problem, for any choice of the 3 masses, there are 3 Euler configurations (also known as the 3 Euler points). These small upper bounds strengthen the belief of Kushnirenko and Khovanskii [18]: real varieties defined by simple systems should have a simple topology. We indicate some hard conjectures about the configurations of relative equilibrium and suggest they could be attacked within the quasipolynomial framework.

Albouy A.
On a paper of Moeckel on central configurations
2003, vol. 8, no. 2, pp. 133142
Abstract
This paper is devoted to general properties of the central configurations: we do not make any restriction on the number $n\geqslant$3 of particles nor on the dimension $d\geqslant1$ of the configuration. Part 7 considers however the particular case $n=d+2$, of interest because the case $n=4$, $d=2$ is the first for which we cannot solve the equations for central configurations. Our main result is Proposition 6, which gives some estimates implying an important estimate due to [23]. Our main tool is Equation $(4.5)$.

Abdullah K., Albouy A.
On a Strange Resonance Noticed by M.Herman
2001, vol. 6, no. 4, pp. 421432
Abstract
The linearized averaged system is a classical integrable approximation for a system of n planets. It presents a strange kind of degeneracy that we call "Herman's resonance". The unexpected identity in the expansion of the perturbating function that causes this degeneracy is shown to be part of a family of similar identities. We give a simple Lemma putting together these identities. We also give a precise construction of the averaged system, that allows us to discuss its intrinsic character.
